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Answer :
Sure! Let's go through each part and simplify the given rational numbers one by one.
a. Simplify [tex]\(\frac{32}{40}\)[/tex]:
- We need to find the greatest common divisor (GCD) of 32 and 40, which is 8.
- Divide both the numerator and the denominator by 8: [tex]\(\frac{32 \div 8}{40 \div 8} = \frac{4}{5}\)[/tex].
- So, [tex]\(\frac{32}{40}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex].
b. Simplify [tex]\(\frac{45}{117}\)[/tex]:
- Find the GCD of 45 and 117, which is 9.
- Divide both the numerator and the denominator by 9: [tex]\(\frac{45 \div 9}{117 \div 9} = \frac{5}{13}\)[/tex].
- Thus, [tex]\(\frac{45}{117}\)[/tex] simplifies to [tex]\(\frac{5}{13}\)[/tex].
c. Simplify [tex]\(\frac{8^2}{9^2}\)[/tex]:
- Calculate [tex]\(8^2 = 64\)[/tex] and [tex]\(9^2 = 81\)[/tex].
- The fraction is [tex]\(\frac{64}{81}\)[/tex]. These numbers have no common factors other than 1, so the fraction is already in its simplest form.
- Therefore, [tex]\(\frac{8^2}{9^2}\)[/tex] is [tex]\(\frac{64}{81}\)[/tex].
d. Simplify [tex]\(\frac{m^2 + m}{1 + m}\)[/tex]:
- Notice that [tex]\(m^2 + m = m(m + 1)\)[/tex].
- Rewrite the fraction as [tex]\(\frac{m(m + 1)}{1 + m}\)[/tex].
- [tex]\((1 + m)\)[/tex] is the same as [tex]\((m + 1)\)[/tex], so it cancels out with the numerator:[tex]\(\frac{m(m + 1)}{m + 1} = m\)[/tex].
- The simplified form is [tex]\(m\)[/tex].
e. Simplify [tex]\(\frac{m^2 + 1}{m + 1}\)[/tex]:
- In this case, [tex]\(m^2 + 1\)[/tex] does not factor to include [tex]\((m + 1)\)[/tex].
- Therefore, we cannot simplify it any further using integer coefficients.
- So, the expression remains [tex]\(\frac{m^2 + 1}{m + 1}\)[/tex].
f. Simplify [tex]\(\frac{m^2 - n^2}{m - n}\)[/tex]:
- Recognize this as a difference of squares: [tex]\(m^2 - n^2 = (m + n)(m - n)\)[/tex].
- The fraction [tex]\(\frac{(m + n)(m - n)}{m - n}\)[/tex] cancels the [tex]\((m - n)\)[/tex] terms.
- This leaves us with [tex]\(m + n\)[/tex].
In summary, the simplified forms are:
- a. [tex]\(\frac{4}{5}\)[/tex]
- b. [tex]\(\frac{5}{13}\)[/tex]
- c. [tex]\(\frac{64}{81}\)[/tex]
- d. [tex]\(m\)[/tex]
- e. [tex]\(\frac{m^2 + 1}{m + 1}\)[/tex]
- f. [tex]\(m + n\)[/tex]
a. Simplify [tex]\(\frac{32}{40}\)[/tex]:
- We need to find the greatest common divisor (GCD) of 32 and 40, which is 8.
- Divide both the numerator and the denominator by 8: [tex]\(\frac{32 \div 8}{40 \div 8} = \frac{4}{5}\)[/tex].
- So, [tex]\(\frac{32}{40}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex].
b. Simplify [tex]\(\frac{45}{117}\)[/tex]:
- Find the GCD of 45 and 117, which is 9.
- Divide both the numerator and the denominator by 9: [tex]\(\frac{45 \div 9}{117 \div 9} = \frac{5}{13}\)[/tex].
- Thus, [tex]\(\frac{45}{117}\)[/tex] simplifies to [tex]\(\frac{5}{13}\)[/tex].
c. Simplify [tex]\(\frac{8^2}{9^2}\)[/tex]:
- Calculate [tex]\(8^2 = 64\)[/tex] and [tex]\(9^2 = 81\)[/tex].
- The fraction is [tex]\(\frac{64}{81}\)[/tex]. These numbers have no common factors other than 1, so the fraction is already in its simplest form.
- Therefore, [tex]\(\frac{8^2}{9^2}\)[/tex] is [tex]\(\frac{64}{81}\)[/tex].
d. Simplify [tex]\(\frac{m^2 + m}{1 + m}\)[/tex]:
- Notice that [tex]\(m^2 + m = m(m + 1)\)[/tex].
- Rewrite the fraction as [tex]\(\frac{m(m + 1)}{1 + m}\)[/tex].
- [tex]\((1 + m)\)[/tex] is the same as [tex]\((m + 1)\)[/tex], so it cancels out with the numerator:[tex]\(\frac{m(m + 1)}{m + 1} = m\)[/tex].
- The simplified form is [tex]\(m\)[/tex].
e. Simplify [tex]\(\frac{m^2 + 1}{m + 1}\)[/tex]:
- In this case, [tex]\(m^2 + 1\)[/tex] does not factor to include [tex]\((m + 1)\)[/tex].
- Therefore, we cannot simplify it any further using integer coefficients.
- So, the expression remains [tex]\(\frac{m^2 + 1}{m + 1}\)[/tex].
f. Simplify [tex]\(\frac{m^2 - n^2}{m - n}\)[/tex]:
- Recognize this as a difference of squares: [tex]\(m^2 - n^2 = (m + n)(m - n)\)[/tex].
- The fraction [tex]\(\frac{(m + n)(m - n)}{m - n}\)[/tex] cancels the [tex]\((m - n)\)[/tex] terms.
- This leaves us with [tex]\(m + n\)[/tex].
In summary, the simplified forms are:
- a. [tex]\(\frac{4}{5}\)[/tex]
- b. [tex]\(\frac{5}{13}\)[/tex]
- c. [tex]\(\frac{64}{81}\)[/tex]
- d. [tex]\(m\)[/tex]
- e. [tex]\(\frac{m^2 + 1}{m + 1}\)[/tex]
- f. [tex]\(m + n\)[/tex]
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